Properties

Label 9075.2.a.db
Level $9075$
Weight $2$
Character orbit 9075.a
Self dual yes
Analytic conductor $72.464$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.4642398343\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{11})\)
Defining polynomial: \(x^{4} - 7 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + q^{3} + ( 2 + \beta_{3} ) q^{4} + \beta_{1} q^{6} + ( \beta_{1} - \beta_{2} ) q^{7} + ( \beta_{1} + 2 \beta_{2} ) q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + q^{3} + ( 2 + \beta_{3} ) q^{4} + \beta_{1} q^{6} + ( \beta_{1} - \beta_{2} ) q^{7} + ( \beta_{1} + 2 \beta_{2} ) q^{8} + q^{9} + ( 2 + \beta_{3} ) q^{12} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{13} + 2 q^{14} + ( 4 + \beta_{3} ) q^{16} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{17} + \beta_{1} q^{18} + ( \beta_{1} - 4 \beta_{2} ) q^{19} + ( \beta_{1} - \beta_{2} ) q^{21} -8 q^{23} + ( \beta_{1} + 2 \beta_{2} ) q^{24} + ( 2 - \beta_{3} ) q^{26} + q^{27} + 2 \beta_{2} q^{28} + 4 \beta_{1} q^{29} + \beta_{3} q^{31} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{32} + ( 12 + 2 \beta_{3} ) q^{34} + ( 2 + \beta_{3} ) q^{36} -5 q^{37} + ( -4 - 3 \beta_{3} ) q^{38} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{39} + 4 \beta_{2} q^{41} + 2 q^{42} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{43} -8 \beta_{1} q^{46} + ( 4 + 2 \beta_{3} ) q^{47} + ( 4 + \beta_{3} ) q^{48} + ( -4 - \beta_{3} ) q^{49} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{51} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{52} + ( 6 - 2 \beta_{3} ) q^{53} + \beta_{1} q^{54} + 2 \beta_{3} q^{56} + ( \beta_{1} - 4 \beta_{2} ) q^{57} + ( 16 + 4 \beta_{3} ) q^{58} + 4 q^{59} + ( \beta_{1} + 2 \beta_{2} ) q^{61} + ( \beta_{1} + 2 \beta_{2} ) q^{62} + ( \beta_{1} - \beta_{2} ) q^{63} -\beta_{3} q^{64} + ( 4 + 3 \beta_{3} ) q^{67} + ( 6 \beta_{1} + 8 \beta_{2} ) q^{68} -8 q^{69} + ( 6 + 2 \beta_{3} ) q^{71} + ( \beta_{1} + 2 \beta_{2} ) q^{72} -3 \beta_{2} q^{73} -5 \beta_{1} q^{74} + ( -9 \beta_{1} + 2 \beta_{2} ) q^{76} + ( 2 - \beta_{3} ) q^{78} + ( -2 \beta_{1} - \beta_{2} ) q^{79} + q^{81} + ( 8 + 4 \beta_{3} ) q^{82} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{83} + 2 \beta_{2} q^{84} + ( 18 + 3 \beta_{3} ) q^{86} + 4 \beta_{1} q^{87} + ( -4 + 4 \beta_{3} ) q^{89} + ( 7 - 3 \beta_{3} ) q^{91} + ( -16 - 8 \beta_{3} ) q^{92} + \beta_{3} q^{93} + ( 6 \beta_{1} + 4 \beta_{2} ) q^{94} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{96} + ( 7 + \beta_{3} ) q^{97} + ( -5 \beta_{1} - 2 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 6 q^{4} + 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{3} + 6 q^{4} + 4 q^{9} + 6 q^{12} + 8 q^{14} + 14 q^{16} - 32 q^{23} + 10 q^{26} + 4 q^{27} - 2 q^{31} + 44 q^{34} + 6 q^{36} - 20 q^{37} - 10 q^{38} + 8 q^{42} + 12 q^{47} + 14 q^{48} - 14 q^{49} + 28 q^{53} - 4 q^{56} + 56 q^{58} + 16 q^{59} + 2 q^{64} + 10 q^{67} - 32 q^{69} + 20 q^{71} + 10 q^{78} + 4 q^{81} + 24 q^{82} + 66 q^{86} - 24 q^{89} + 34 q^{91} - 48 q^{92} - 2 q^{93} + 26 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 7 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 5 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 4\)
\(\nu^{3}\)\(=\)\(2 \beta_{2} + 5 \beta_{1}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.52434
−0.792287
0.792287
2.52434
−2.52434 1.00000 4.37228 0 −2.52434 −0.792287 −5.98844 1.00000 0
1.2 −0.792287 1.00000 −1.37228 0 −0.792287 −2.52434 2.67181 1.00000 0
1.3 0.792287 1.00000 −1.37228 0 0.792287 2.52434 −2.67181 1.00000 0
1.4 2.52434 1.00000 4.37228 0 2.52434 0.792287 5.98844 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9075.2.a.db yes 4
5.b even 2 1 9075.2.a.cw 4
11.b odd 2 1 inner 9075.2.a.db yes 4
55.d odd 2 1 9075.2.a.cw 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9075.2.a.cw 4 5.b even 2 1
9075.2.a.cw 4 55.d odd 2 1
9075.2.a.db yes 4 1.a even 1 1 trivial
9075.2.a.db yes 4 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):

\( T_{2}^{4} - 7 T_{2}^{2} + 4 \)
\( T_{7}^{4} - 7 T_{7}^{2} + 4 \)
\( T_{13}^{4} - 46 T_{13}^{2} + 1 \)
\( T_{17}^{2} - 44 \)
\( T_{19}^{4} - 79 T_{19}^{2} + 1156 \)
\( T_{23} + 8 \)
\( T_{37} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 7 T^{2} + T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( T^{4} \)
$7$ \( 4 - 7 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( 1 - 46 T^{2} + T^{4} \)
$17$ \( ( -44 + T^{2} )^{2} \)
$19$ \( 1156 - 79 T^{2} + T^{4} \)
$23$ \( ( 8 + T )^{4} \)
$29$ \( 1024 - 112 T^{2} + T^{4} \)
$31$ \( ( -8 + T + T^{2} )^{2} \)
$37$ \( ( 5 + T )^{4} \)
$41$ \( ( -48 + T^{2} )^{2} \)
$43$ \( ( -99 + T^{2} )^{2} \)
$47$ \( ( -24 - 6 T + T^{2} )^{2} \)
$53$ \( ( 16 - 14 T + T^{2} )^{2} \)
$59$ \( ( -4 + T )^{4} \)
$61$ \( 256 - 43 T^{2} + T^{4} \)
$67$ \( ( -68 - 5 T + T^{2} )^{2} \)
$71$ \( ( -8 - 10 T + T^{2} )^{2} \)
$73$ \( ( -27 + T^{2} )^{2} \)
$79$ \( 1 - 46 T^{2} + T^{4} \)
$83$ \( 256 - 76 T^{2} + T^{4} \)
$89$ \( ( -96 + 12 T + T^{2} )^{2} \)
$97$ \( ( 34 - 13 T + T^{2} )^{2} \)
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